With his master, Darth Maul hoped to destroy the Jedi and restore the Sith to galactic power. However, Maul's hatred of the Jedi was more personal. He was filled with anger when he realized that as a child, the Jedi had not come for him.
He hates Obi-Wan, he fears Palpatine. Based on his reactions to Palpatine and Vader, he's terrified of the Sith. Obi-Wan. The thing is, Maul hates Palpatine but he has respect for how powerful he is and his plans.
Regardless, based on the season 7 of The Clone Wars, and the fact that Darth Maul was aware of Anakin Skywalker's future and his role as the Chosen One, it is very likely that Darth Maul indeed feared Darth Vader to some extent, and preferred to avoid a direct confrontation with him, at least, on his own.
Maul may have developed a slight fondness for Ezra and still used Ezra to achieve his own goals, it did appear that Maul genuinely wanted Ezra as his apprentice. However, it is made clear by Sam Witwer that Maul does want Ezra as a replacement for his dead brother.
Training in the ways of the Force under the mentorship of Kanan Jarrus, Ezra became a Jedi Padawan and a part of the emerging rebellion. He also grew close to Hera Syndulla, Zeb, Chopper, and Sabine Wren, who was like a sister in his new found family.
Star Wars Rebels lead creator Dave Filoni even admitted that Maul and Vader were meant to have a fight, but he felt it would take away from the battle with Ahsoka. But as things stand, Star Wars not only lacks a fight between Vader and Maul, but the two have never even spoken to each other.
Maul's skills, on the other hand, became stagnant as he spent the last few decades of his life keeping himself alive and plotting schemes. Given this, it makes sense Obi-Wan quickly defeated Maul in their final duel in Star Wars Rebels.
In the film's climax, Maul duels Qui-Gon and his apprentice Obi-Wan Kenobi (Ewan McGregor). He is able to kill Qui-Gon and knock Obi-Wan down a shaft, but Kenobi propels himself out of the pit and uses Qui-Gon's lightsaber to bisect Maul, sending the two parts of his body falling down the shaft.
During the battle, Anakin, who had been known as the Sith Lord Darth Vader, was redeemed by Luke and brought balance to the Force. However, the redemption cost Anakin his life, having been mortally wounded by the Emperor, Darth Sidious, while killing his former Master. After his death, Anakin became one with the Force.
Most of the Hutt Council escaped, though Maul tracked them down to Jabba the Hutt's palace on Tatooine. There, he had Jabba the Hutt swear fealty to the Shadow Collective. After a staged attack on Mandalore positioning Death Watch as saviors, Pre Vizsla betrayed Maul and assumed total control of the planet.
The fight was orchestrated by a darksider cult called the Prophets of the Dark Side, who wanted Vader killed and replaced with Maul as Darth Sidious' only true apprentice, because they felt the former was "tainted" by his Jedi past and not worthy of being a Sith.
Like his master, Darth Sidious, better known as Sheev Palpatine, Maul's main weakness was his overconfidence. In Luke Skywalker's confrontation with Palpatine during The Return of the Jedi, he tells the Emperor that his overconfidence is his weakness.
By the time of Episode 3; Anakin is in his prime, and already one of the Jedi Order's most powerful “masters” (although technically still a knight), just behind Obi-Wan, Windu, and Yoda. Maul wouldn't stand a chance.
In the end, Vader reigns supreme against the reborn villain as his hatred drives him to be named the winner of the fight. In the end, Darth Vader's inner hatred and darkness made him the winner of the epic battle against Darth Maul.
Darth Maul could have killed Anakin Skywalker at least twice and therefore prevented Darth Vader from being created, but that does not mean he would have been powerful enough to stop Darth Sidious' rise to power.
Had they fought one another at their full potential, Maul's exceptional lightsaber skills would have been difficult for Dooku, but Dooku's knowledge of the dark side of the Force would likely have prevailed.
In Tales of the Jedi, set thousands of years before the original Star Wars film, the Mandalorians are a major military power who side with the Sith in their war against the Jedi, and their leader is manipulated by the Sith into triggering a war with the Republic.
In canon Luke has never turned to The Dark-Side so he has no Sith name there. In Legends, Luke does turn to the Dark-Side when Darth Sidious reemerges in a new clone body. Luke never became a full Dark Lord of the Sith though before being brought back to the Light, therefore he has no Sith name in Legends either.
